I agree with Morten. If you are referencing service as in product support and help, I focus on net promoter score (NPS) first and foremost - is your product and are your people easy to work with? If so, customers will promote you (sell for you) in the marketplace. Your customers are your best product development resource, they will tell you what works and what they need more/less of. Support people need to be empowered to solve issues in the moment so you can learn. Case ratios, resolution, response time, customer satisfaction are all important but how you measure/manage changes based on stage of growth and maturity. As you grow you have to focus on scale and how you can solve issues in product or self serve.
If you are referencing customer success - the focus on adoption and utilization of your product, then I have four core measures, adoption/utilization, NPS, logo churn and revenue growth. Make those measures part of a quarterly MBO. A big mistake is to hire someone that is not familiar, too junior or lacking in business experience relative to your end users. Your customers need to learn from you, not teach your customer success team. Hire someone (like a customer user who has implemented your product) so they can immediately add value to your customers.